How to split one control end of bezier curve
I think you can create such shapes only by combining 2 or multiple curves, because 


  Each new segment is added to one end of the curve. A new segment will
  only be added if a single vertex, or handle, at one end of the curve
  is selected.


wiki page

The same may be applied to creating 'P' shape, thus joining 2 curve segments - it'll raise an error "Cannot make segment". You can duplicate that curve segment and place it atop of existing one, but still it'd be 2 different curves (example below).

As another way you may want to use simple mesh with all vertices deleted except for one. You then extrude shape needed either with E or Ctrl + LMB:



After adding mirror modifier you'll get that 'Y' shape. If you convert now this mesh into curve with Alt+C you'll see that it actually consists of 2 curves:



It's just the same what you would get if extruding this shape with curves manually.